Quantum annealing in a kinetically constrained system

Classical and quantum annealing is discussed in the case of a generalized kinetically constrained model, where the relaxation dynamics of a system with trivial ground state is retarded by the appearance of energy barriers in the relaxation path, following a local kinetic rule. Effectiveness of thermal and quantum fluctuations in overcoming these kinetic barriers to reach the ground state are studied. It has been shown that for certain barrier characteristics, quantum annealing might by far surpass its thermal counter part in reaching the ground state faster.

Figure 1

Potential energy wells for the spin at site $i$, when $(i-1)$-th spin is (a) up and (b) down, with the external field $h$ in the downward direction and barrier height $\chi$ very large and the width $a$ small. For the classical generalized East model, flipping across the barrier in (b) is a thermal jump (at any finite $T$). In the quantum model considered here, probability for crossing the barrier in (b) is due to quantum tunneling through it at a finite $\Gamma$.

Figure 2

Quantum annealing $(T=0)$ for $g=100$, ${\Gamma }_0=100$, and $h=1$ is shown for different values of ${\tau }_Q$, for a chain of $5\times {10}^4$ spins ($m$ averaged over the same set of ten initial configurations for each ${\tau }_Q$). The horizontal (dashed) line indicates the average (over the same set $\mathcal{C}$) value of $m$ that could be reached from the initial configurations by simply minimizing the energy following the downhill principle (a single step is enough to get there). In the inset, variation of $\ln \left[{\left({\tau }_Q\right)}_{\min }\right]$ with $\ln \left[g\right]$ is shown (by the points) for one given configuration. The error in ${\left({\tau }_Q\right)}_{\min }$ is typically less than 0.5% in each case. The continuous line in the inset shows a fit of the data by the continuous line ${\left({\tau }_Q\right)}_{\min }\sim g^{\kappa }$; $\kappa \approx 1.67$ (obtained by linear least-square fitting).

Figure 3

Comparison between classical and quantum annealing for a chain of $5\times {10}^4$ spins (for the same initial disordered configuration with $m_i\sim {10}^{-3}$). We show the results for ${\tau }_Q=1.8\times {10}^3$ (for quantum) and ${\tau }_C={10}^6$ (for classical) with $h=1$; a lower ${\tau }_C$ would not produce substantial annealing. Starting from the same initial values ${\Gamma }_0=T_0=100$, (and $g=100$ in the quantum case) we observe that classical annealing requires about ${10}^7$ steps, whereas quantum annealing takes about ${10}^4$ steps for achieving the same final order $m_f\sim 0.92$.